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Use the differentiation rules developed in this section to find the derivatives of the functions in Exercises 35-64. Note that it may be necessary to do some preliminary algebra before differentiating.

f(x)=x-1(x+1)(x+2)

Short Answer

Expert verified

The derivative of the function isf'(x)=-x2+2x+5(x2+3x+2)2

Step by step solution

01

Step 1. Given Information

The given function isf(x)=x-1(x+1)(x+2).

02

Step 2. Simplify the function

Simplify the denominator of the function.

f(x)=x-1x(x+2)+1(x+2)=x-1x2+2x+x+2=x-1x2+3x+2

03

Step 3. Find the derivative

Apply the quotient rule of derivative, (fg)'(x)=f'(x)g(x)-f(x)g'(x)(g(x))2.

f'(x)=ddx(x-1)·(x2+3x+2)-(x-1)ddx(x2+3x+2)(x2+3x+2)2=(1)·(x2+3x+2)-(x-1)(2x+3)(x2+3x+2)2=x2+3x+2-(2x2+3x-2x-3)(x2+3x+2)2=-x2+2x+5(x2+3x+2)2

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